Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(6,11)(7,16)(8,12)(9,20)(13,28)(14,31)(15,25)(17,19)(18,26)(21,27)(22,29)(23,24)(30,34)(32,33);(1,3,4,2)(5,6,12,24,9,14,21,25,10,15,27,31,20,23,8,11)(7,13,26,35,18,28,16,29,19,30,33,36,32,34,17,22);(5,7,8,17,20,32,27,33,10,19,21,16,9,18,12,26)(6,13,11,22,23,34,31,36,15,30,25,29,14,28,24,35)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13144, #494896, #494897, … |
| maximal normal subgroup of | C2×#13144, #494896, #494897, … |
| maximal quotient of | #140172, #140197, #140231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6132);